# Formula question

This isn't a homework question, but I'm just wondering how the text in the book gets from one formula to the next.

To figure out total kinetic energy of a rotating object:

K translational + K rotational

$$\frac{1}{2}mv^2+\frac{1}{2}I (\frac{v}{r})^2$$

Then the book gives an alternate formula:

$$\frac{1}{2}mv^2(1+\frac{I}{mr^2})$$

So I wanted to see how they get from one to the other. So I tried

$$\frac{1}{2}mv^2+\frac{1}{2}I (\frac{v}{r})^2$$

factor out the 1/2v^2

$$\frac{1}{2}v^2 (m+\frac{I}{r^2})$$

almost there. But how do I get a 1 in place of the m? I could divide by m, but then I get

$$\frac{v^2}{2m} (1+\frac{I}{mr^2})$$

It works for the right term, but not the left. What am I doing wrong?

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Fermat
Homework Helper
You just didn't factor out completely.

You ended up with,

$$\frac{1}{2}v^2 (m+\frac{I}{r^2})$$

now take out m. i.e. divide both terms inside the brackets by m so that you end up with m outside the barckets.

Alternatively, you could multiply out the alternate formula, and you will end up with the original one.